Author manuscript, published in "Mechanical Systems and Signal Processing 21, 2 (2007) 1127-1142"
DOI : 10.1016/j.ymssp.2006.03.002
A New Approach to Detect Broken Rotor Bars in Induction Machines
by Current Spectrum Analysis
G. Didier∗, E. Ternisien†, O. Caspary∗ , and H. Razik†
Abstract
hal-00115405, version 1 - 6 Dec 2006
In this paper, a new technique to detect broken rotor bars in polyphase induction machines is presented.
Like most techniques, we employ the Fourier Transform of one stator current to make detection. But where
the other methods use the Fourier Transform modulus, we propose an alternative approach by analyzing
its phase. As shown by results, the Fourier Transform phase allows to detect one broken rotor bar when
the motor operates under a low load. In order to improve the diagnosis and to permit the detection of
incipient broken rotor bar, we complete the analysis with the Hilbert Transform. This transform provides
good results and a partially broken rotor bar can be detected when the load torque is equal or greater than
25%. The main advantage of these methods is that it does not require a healthy motor reference to take the
final decision on the rotor cage state.
keywords : Phase analysis, Fourier Transform, Hilbert Transform, Fault diagnosis, Induction machines.
1
Introduction
Nowadays, no one can deny the important role of the asynchronous motor in industry applications. It is wellknown that an interruption of a manufacturing process due to a mechanical or electrical problem induces a
significant financial loss for the firm. Interruptions can be caused by rotor faults (broken rotor bars or cracked
rotor end ring), stator faults (opening of a stator phase or inter-turn short circuits) , rotor-stator eccentricity
(static and dynamic eccentricity) and bearing failures [1], [2]. In order to avoid such problems, we have to detect
these faults to prevent a major failure from occurring.
Broken rotor bars rarely cause immediate failures, especially in large multi-pole (slow speed) motor. However,
if there are enough broken rotor bars, the motor may not start as it may not be able to develop sufficient
accelerating torque. Regardless, the presence of broken rotor bars precipitates deterioration in other components
that can result in time-consuming and expensive fixes.
Various techniques have been developed to detect broken rotor bars in induction motors. We can quote
vibration measurement [3], temperature measurement [4], coils to monitor the motor axial flux [5], Park’s Vector
currents monitoring [6], artificial intelligence based techniques [7]. However, the most popular techniques are
based on the monitoring of the stator current spectrum (called Motor Current Signature Analysis) because of
its non-intrusive feature [8, 9, 10, 11]. In this technique, the amplitude of the lateral bands created by the rotor
fault around the supply frequency are monitored. An augmentation of these amplitudes allows dimensioning
the failure’s degree. Others use the instantaneous power spectrum of one stator phase to calculate a global fault
index [12]. The disadvantage of all these methods is that the knowledge of the healthy motor stator current is
necessary to take a decision about the rotor state.
In this paper, we propose a broken rotor bar detection method using the line current Discrete Fourier
Transform (DFT) phase. This technique does not require the healthy motor current knowledge, which is a
major advantage compared to the classical ones. We will show that the basically calculated phase gives good
results when the motor operates near its nominal load. For weak load, the results obtained are not robust
enough especially for the detection of an incipient rotor fault. This problem will be solved by using the Hilbert
Transform applied to the line current spectrum modulus. Thanks to this method, the diagnosis of a partially
broken rotor bar could be carried out without reference even if the motor operates at low load (25% of the rated
torque).
∗ Groupe
de Recherche en Electrotechnique et Electronique de Nancy, GREEN-CNRS UMR-7037, Université Henri Poincaré,
54506 Vandœuvre-lès-Nancy Cedex, France. e-mail: gaetan.didier@green.uhp-nancy.fr;hubert.razik@green.uhp-nancy.fr
† Centre de Recherche en Automatique de Nancy, CRAN-CNRS UMR-7039, Université Henri Poincaré, 54506 Vandœuvre-lèsNancy Cedex, France. e-mail: eric.ternisien@iutsd.uhp-nancy.fr; olivier.caspary@iutsd.uhp-nancy.fr
1
2
Current monitoring
Consider an ideal three-phase supply and an asynchronous motor connected in wye. Thus the instantaneous
current circulating in one phase is defined as:
√
is0 (t) = 2Is sin(ωs t − ϕ)
(1)
with regard to the instantaneous voltage:
v(t) =
√
2Vs sin(ωs t)
(2)
The term ϕ represents the phase angle between the voltage and the line current. We can see in Eq. (1) that
the power spectrum of the current will contain only one fundamental component at frequency fs = ωs /2π.
When one bar breaks, a rotor asymmetry is thus created. The result is the appearance of a backward
rotating field at the slip frequency sfs (s represents the slip and fs the supply frequency). This induces, in the
−
= (1 − 2s)fs . This cyclic current variation
stator current spectrum, an additional component at frequency fbb
1
causes a speed oscillation at twice the slip frequency 2sfs [14]. This speed oscillation induces, in the stator
+
= (1 + 2s)fs , and so on. By extension, the broken rotor bar
current spectrum, an upper component at fbb
1
creates additional components in the spectrum modulus at frequencies given by [14]:
k = 1, 2, 3, . . .
(3)
The effects of a broken rotor bar can be seen in Fig. 1. We represent the line current spectrum in the case
of a healthy rotor in Fig. 1(a), and with one broken rotor bar in Fig. 1(b). We can see that the appearance
±
of a rotor fault increase the amplitude of the component situated at frequencies fbb
. The amplitude of these
k
frequencies are dependent of three factors. The first is the motor’s load inertia, the second is the motor’s load
torque (current in the rotor bars) and the third is the severity of the rotor fault. For example, the magnitude
of these components will be more important with three broken bars than with a fissured one. If we analyze
−
= (1 − 2s)fs . This
Fig. 1(a), we notice that the current spectrum contains a component at the frequency fbb
1
frequency is created by the asymmetry existing in all induction motors and is usually used as reference for the
rotor fault diagnosis.
0
−20
−20
(1 − 2ks)fs
(1 + 2ks)fs
−40
(1 + 2ks)fs
−40
−60
−60
−80
−100
10
(1 − 2ks)fs
Relative Power Spectral Density (dB)
0
Relative Power Spectral Density (dB)
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±
fbb
= (1 ± 2ks)fs
k
−80
20
30
40
50
60
Frequency (Hz)
70
80
−100
10
90
20
30
40
50
60
Frequency (Hz)
70
80
90
(b) One broken bar
(a) Healthy rotor
Figure 1: Line current spectrum modulus for a healthy rotor (a) and a rotor with one broken bar (b)
The classical broken bar detection methods usually use the monitoring of the line current Fourier Transform
modulus. They are based on the appearance or the increase of the components amplitude at frequencies defined
in Eq. (3). Appearance or increase are two terms which induce the comparison with a reference which is often
the Fourier Transform modulus of the current line absorbed with a healthy rotor as we explained previously.
In this work, the spectrum reference given by the line current of the healthy motor is not necessary anymore.
Thus, we propose to detect a broken rotor bar without a priori knowledge of the motor state.
2
4
4
3
3
(1 − 2ks)fs
1
Phase (Rad)
Phase (Rad)
2
0
(1 + 2ks)fs
1
0
−1
−1
−2
−2
−3
−3
−4
10
(1 − 2ks)fs
2
20
30
40
50
60
Frequency (Hz)
70
80
−4
10
90
(1 + 2ks)fs
20
30
40
50
60
Frequency (Hz)
70
80
90
(b) One broken bar
(a) Healthy rotor
Figure 2: Line current spectrum phase for a healthy rotor (a) and a rotor with one broken bar (b)
hal-00115405, version 1 - 6 Dec 2006
3
Discrete Fourier Transform (DFT) phase analysis
The DFT is naturally obtained by a discretisation at the sampling frequency of the Discrete Time Fourier
Transform (DTFT) given by:
N
−1
X
X(f ) =
x(n)e−2jπnf
(4)
n=0
In the case where x(n) = e2jπnf0 , the DTFT of x(n) is written:
X(f ) =
N
−1
X
e2jπnf0 e−2jπnf =
n=0
sin(N π(f − f0 )) −j2π(N −1)(f −f0 )
e
sin(π(f − f0 ))
(5)
πf ) −j2π(N −1)f
In this expression, we find the Dirichlet function sin(N
introduced by the DTFT of the rectangular
sin(πf ) e
window.
Let us consider the following line current expression:
Kl √
X
2Is m−
ck
is (t) = is0 (t) +
cos(2π(fs − k ff )t − ϕ−
(6)
k)
2
k=1
Kr √
X
2Is m+
ck
cos(2π(fs + k ff ) t − ϕ+
+
k)
2
k=1
+
where m−
ck and mck represent the modulation index of the left component k and the modulation index of the
right component k with regards to the supply frequency fs in the spectrum modulus. Their values are a function
+
of the fault level present in the rotor cage as we showed in [12]. Terms ϕ−
k and ϕk represent the dephasing
of the component (left or right) according to the origin (0 rad). Terms Kl and Kr respectively represent the
number of components at the left and at the right of the carrier frequency present in the line current spectrum
for a healthy or faulty rotor. The frequency ff represents the modulation frequency introduced by the natural
asymmetry or by the rotor fault (ff = 2sfs ). The corresponding line current spectra are given in Figs. 1.
Classically, the diagnosis methods use the DFT modulus of the line current defined in Eq. (6) to detect a
fault. We have chosen to use the phase because it gives a better representation of the information contained in
±
the line current. Indeed, the components at frequencies fbb
(Eq. (3)) are present, and their amplitudes are a
k
function of the fault level, as we can see in Figs. 2.
In fact, on each side of the 50 Hz, we observe the presence of phase jumps between +π and −π. These phase
jumps are another representation of the components present in the line current spectrum modulus. Furthermore,
in this representation, we can observe a phase jump at the 50 Hz frequency.
The presence of this jump is due to the fact that the frequencies contained in the line current spectrum are
not a multiple integer of the resolution frequency ∆f used (∆f = Fe /N where Fe and N respectively represent
the sampling frequency and the total number of points used for the calculation of the line current spectrum).
3
Indeed, in the case where fs and ff are multiple integers of ∆f (fs = x ∆f and ff = x ∆f with x an
integer) the line current spectrum phase is illustrated in the Fig. 3(a) for ff = 6 Hz (parameters used in this
case are indexed in Table 1). We obtain the dephasing of all sinusoids contained in the line current expression
is (t). However, in the case where the frequencies fs and ff are not multiple integers of ∆f , we obtain the
representation given in Fig. 3(b) (by preserving the same parameters). As we can see in Fig. 3(c), if the
modulation indexes increase, which means the appearance of a defect in the rotor cage, the jumps present at
frequencies (1 ± 2ks)fs increase too (in this case, parameters used are given in Table 2).
Table 1: Parameters used for Figs. 3(a) and 3(b) (Kl = Kr = 2).
k
m ck
ϕk
1−
0.003
−π/4
1+
0.002
π/3
2−
0.0015
−π/7
2+
0.0005
π/8
Table 2: Parameters used for the Fig. 3(c) (Kl = Kr = 2).
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k
m ck
ϕk
1−
0.004
−π/4
1+
0.003
π/3
2−
0.002
−π/7
2+
0.001
π/8
In order to understand why the phase of the line current spectrum takes this form along the frequency axis,
we propose to study a simple signal. The Discrete Time Fourier Transform (DTFT) of the line current spectrum
given at Eq. (6), if we consider only its fundamental component with ϕ = 0 (frequency fs ), can be expressed
as:
Is (f )
=
√
2Is
N
−1
X
sin(2πfs t) e−2jπnf
(7)
n=0
N
−1 j2πfs n
X
√
e
− e−j2πfs n −2jπnf
2Is
e
= |Is (f )|e−ΨIs (f )
2j
n=0
√
2Is sin(N π(f − fs )) −j2π(N −1)(f −fs )
e
=
2j sin(π(f − fs ))
√
2Is sin(N π(f + fs )) −j2π(N −1)(f +fs )
−
e
2j sin(π(f + fs ))
=
In this expression, we find two Dirichlet functions positioned at frequencies −fs and +fs . As said previously,
the DFT is a sampling of the DTFT. So, in the case where fs is not a multiple integer of ∆f , we do not find
a simple component at frequency fs in the line current spectrum modulus but parasitic undulations as shown
in Fig. 4(a). Indeed, instead of sampling the DTFT when it equals 0 (case fs = x ∆f ), the values obtained are
the values of the Dirichlet functions.
The same phenomenon occurs for the real and imaginary parts of the spectrum Is (f ). They are not null
and depend on values taken by the real and imaginary parts of the DTFT of the line current (Fig. 4(b)).
As the phase of the DFT ΨF T (f ) varies according to the sign of the real and imaginary parts of Is (f ) (Eq.
(8)), we obtain the form given in Fig. 4(c) with a positive jump at the frequency fs like in Figs. 2(a) and 2(b).
In the ideal case, we would have obtained a component of value −π/2 located at the frequency fs like in Fig.
3(a).
IF T (I(f ))
(8)
ΨF T (f ) = arctan
RF T (I(f ))
Nevertheless, the phase jump at 50 Hz is very advantageous for the fault detection because we are no longer
disturbed by the fundamental component energy problem present in the spectrum modulus. This is interesting
for the study of high power motors which have a weak slip (for example, the slip of a 2MW induction motor is
near to 0.3%, which gives a frequency ff of 0.3 Hz if the fundamental frequency fs is equal to 50 Hz).
Now, let us consider the particular frequency (1 − 2s)fs . This frequency, in the case of a healthy rotor,
corresponds to the natural rotor eccentricity. So, as shown in Fig. 2(a), there is no phase jump at frequency
4
2
ϕ−
2
1
ϕ−
1
0
−1
ϕ
−2
ϕ+
1
ϕ+
2
−3
30
40
50
Frequency (Hz)
60
3
(1 − 2ks)fs
2
1
0
(1 + 2ks)fs
−1
−2
(1 − 2ks)fs
2
1
0
(1 + 2ks)fs
−2
−3
(a) fs = x ∆f and ff = x ∆f
3
−1
−3
−4
70
Discrete Fourier transform phase (Rad)
3
−4
4
4
Discrete Fourier transform phase (Rad)
Discrete Fourier transform phase (Rad)
4
30
40
50
Frequency (Hz)
60
−4
70
30
(b) fs 6= x ∆f and ff 6= x ∆f
40
50
Frequency (Hz)
60
70
(c) fs 6= x ∆f and ff 6= x ∆f
40
(1)
3.5
Re(DTFT)
Re(DFT)
20
3
0
2.5
−20
−40
0
20
20
40
60
(2)
80
100
Im(DTFT)
Im(DFT)
0
40
60
Frequency (Hz)
80
(a) Modulus of Is (f )
100
−60
0
1
0.5
−0.5
−40
20
2
1.5
0
−20
0
Phase of the DFT (Rad)
Abs(DTFT)
Abs(DFT)
Magnitude (number of samples)
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N/2
Real part (1) and imaginary part (2) of DTFT and DFT
Figure 3: Current spectrum phase of Eq. (6): Parameters of Table 1 (a) and (b) and Parameters of Table 2 (c).
20
40
60
Frequency (Hz)
80
100
−1
0
(b) Real and imaginary parts of Is (f )
20
40
60
Frequency (Hz)
80
100
(c) Phase of Is (f )
Figure 4: Discrete Fourier Transform of line current (Eq. (7)) with fs 6= x FNe .
(1 + 2s)fs because there is no broken rotor bar (in fact a component exists but it is masked by the noise).
That is checked for all load levels studied. On the opposite, when a broken bar is present in the rotor cage
(Fig. 2(b)), we can see that the phase jump at frequency (1 + 2s)fs is present (like the other components at
frequencies given by Eq. (3)).
A simple criterion can be deduced from this observation:
• if there is no phase jump at the frequency (1 + 2s)fs , there is no default;
• if there is a phase jump at the frequency (1 + 2s)fs , there is default.
In order to make a more robust detection and to limit the detection of false alarms, a threshold α has been introduced in the criterion. This threshold compares the variance σc of ΨF T (f ) in the interval Bc = [(1 + 2s)fs − 2δ , (1 + 2s)fs + 2δ ]
where the fault component is located, with the variance σn of ΨF T (f ) in the interval Bn = [(1 + 2s)fs + 2δ , (1 +
4s)fs − 2δ ] which contains noise. The term δ represents the frequency band width used for the calculus of σc
(Fig. 5).
So the criterion can be reformulated as:

σc
>α
⇒ Default
if



σn
(9)


 if σc ≤ α
⇒ No Default
σn
Since the detection is based on the appearance of a significant phase jump at the frequency (1 + 2s)fs , the
first step of the algorithm is to estimate the slip of the induction motor. To do that, we recommend to use the
component at frequency (1 − 2s)fs because, in the case of a healthy rotor, only this component is detectable.
Moreover, the detection is easier at the left of the 50 Hz due to the greatest amplitude of components. Therefore,
the proposed algorithm for the broken bars detection is:
1. Detection of the maxima (or minima, it depends on the phase shape) at the left of the 50 Hz;
5
2. Selection of the 50 Hz nearest maximum ((1 − 2s)fs component);
3. Computation of σc and σn with respect to the (1 + 2s)fs component;
4. Decision.
4
3
(1 + 2g)fs
Phase (Rad)
2
1
0
Bc
Bn
−1
δ
−2
−3
−4
50
(1 + 4g)fs
51
52
53
54
55
Frequency (Hz)
56
57
58
59
60
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Figure 5: Variance calculus intervals on the line current spectrum phase in the frequency band [50 - 60] Hz
In fact, we can limit the frequency band used for the maxima research between [F0 , 50] Hz. The frequency
F0 depends on the maximum slip of the induction motor (at full load). For our motor, the rotation speed at
full load is Ω = 2804rpm, so s = 6.53% and 2sfs = 6.53 Hz. Consequently, we choose F0 = 40 Hz.
Let us note that in order to suppress the little peaks in the signal, a median filter is used. This filter has a
smoother effect which helps the detection of maxima (or minima). The results of this method are presented in
section 5.
To ensure the robustness of the method explained before, and to detect more precisely the incipient rotor
faults, we propose another approach to improve the broken rotor bar diagnosis.
4
Hilbert Transform (HT) phase analysis
As explained in the previous section, the DFT phase, although the method gives good results, has two disadvantages. The first is that noise level is important, which makes the maxima detection and the peaks discrimination
at the right of the 50 Hz (at low load) difficult. The second is that the shape of the phase is not fixed (see Figs.
8 et 10). Indeed, in the case where studied frequencies are different from ∆f , the real and imaginary parts
can take random values. In order to stabilize the phase shape, we must find a solution to control the real and
imaginary part values of the DFT spectrum. In fact, the idea is to obtain a phase always equal to −π/2 at the
left of fs and equal to π/2 at the right of fs , with no variation excepted at the broken bar frequencies. In other
±
and fs .
words, the real part must be null excepted at frequencies fbb
k
The solution is to interpret the Discrete Fourier Transform modulus as the real part of a new signal called
”analytic signal”. Indeed, as we can see in Fig. 6(a), the modulus contains very weak values and dirac impulsions
at studied frequencies. So, the shape of the real part is well-known at each frequency point and the Signal Noise
Ratio (SNR) is increased.
The analytic signal is obtained by a Hilbert Transform of the line current spectrum modulus. If we consider
a time signal s(t), the analytic signal s̃(t) can be expressed as:
s̃(t) = s(t) + jρ(t).
where
(10)
1
(11)
πt
The HT H(s(t)) is defined as the convolution product of the signal s(t) (which becomes the real part of s̃(t))
1
. The analytic signal s̃(t) is calculated by using different
by a filter whose the impulse response is h(t) = πt
ρ(t) = H(s(t)) = s(t) ∗
6
methods [13]. One of these methods is the use of the Fourier Transform. Indeed, the Fourier Transform of the
signal ρ(t), which represents the Hilbert Transform of the signal s(t), is given by the following equation:
F
ρ(t) −−−−→ −jsgn(f )S(f )
(12)
where the function sgn(f ) and S(f ) represent respectively the signum function (distribution) given by Eq. (13)
and the Fourier Transform of s(t).

 +1 for f > 0
0 for f = 0
(13)
sgn(f ) =

−1 for f < 0
Consequently, the analytic signal s̃(t) can be obtained by using the Fourier Transform with the expression:
F
s̃(t) −−−−→ S(f ) + j[−j sgn(f )]S(f ) = [1 + sgn(f )]S(f ) = x(f )S(f )
hal-00115405, version 1 - 6 Dec 2006
with

 2
1
x(f ) = [1 + sgn(f )] =

0
for f > 0
for f = 0
for f < 0
(14)
(15)
The Fourier image of the analytic signal is doubled at positive frequencies and cancelled at negative frequencies
with respect to S(f ). The advantage of this transformation is that the result remains in the same domain as
the signal analyzed (time for example). The particularity of this paper is that we do not apply the HT on the
time signal is (t) but on its spectrum modulus |Is (f )|. Hence, the analytic signal of this modulus provides:
Ies (f ) = |Is (f )| + j IHT (f )
with
IHT (f ) = H(|Is (f )|)
(16)
(17)
We represent in Eq. (18) the diagram used to obtain the analytic signal Ies (f ) with the Fourier Transform.
F
|Is (f )| −−−−→
x
 e
R(Is (f ))
S(z)

. x(z)
y
F
Ies (f ) ←−−−− x(z)S(z)

 e
yI(Is (f ))
−1
(18)
IHT (f )
The analytic signal phase ΨHT (f ) can be calculated with the expression:
!
I(Ies (f ))
IHT (f )
ΨHT (f ) = arctan
= arctan
|Is (f )|
R(Ies (f ))
(19)
The real part |Is (f )|, the imaginary part IHT (f ) and the phase ΨHT (f ) of the analytic signal are shown in
Figs. 6 in case of one broken rotor bar.
the Hilbert Transform applied to the spectrum modulus gives a phase restricted to the interval
πIn πconclusion,
− 2 , 2 . Moreover, the knowledge of the imaginary part form Eq. (11) allows to predict the exact form of the
analytic signal phase.
Thanks to the noise reduction, the phase jumps are more pronounced which permits an easier detection.
The noise level is lower than in ΨF T (f ) because of the redefinition of |Is (f )| and IHT (f ) thanks to the Hilbert
Transform. The algorithm is the same as for the DFT phase detection.
5
Experimental results
The test-bed used in the experimental investigation is composed of one three phase induction motor, 50 Hz, 2poles, 3kW. In order to test the effectiveness of the suggested methods, several identical rotors can be exchanged
without affecting the electrical and magnetic features. The squirrel cage has 28 rotor bars (Figs. 7). The voltage
7
4
1
x 10
2
0.8
8000
6000
(1 − 2ks)fs
4000
(1 + 2ks)fs
2000
0.6
1.5
Phase of the Analytic Signal
Imaginary Part of the Analytic Signal
Real part of the Analytic Signal (FT Modulus)
10000
(1 − 2ks)fs
0.4
0.2
0
−0.2
−0.4
(1 + 2ks)fs
20
30
40
50
60
Frequency (Hz)
70
80
90
(a)
1
0.5
0
−0.5
(1 + 2ks)fs
−1
−0.6
−1.5
−0.8
0
10
(1 − 2ks)fs
−1
10
20
30
40
50
60
Frequency (Hz)
70
80
90
−2
10
(b)
20
30
40
50
60
Frequency (Hz)
70
80
90
(c)
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Figure 6: Real part (a), imaginary part (b) and phase (c) of the analytic signal I˜s (t) (for one broken rotor bar).
(a)
(b)
Figure 7: Test-bed and rotor with one broken bar
and the line current measurements were made at the nominal rate. For those two variables, the sampling
frequency Fe was 2k Hz and each data length was equal to 218 values. We decided to study the case of a
partially broken rotor bar (approximately 50% of the bar is bored) and one broken rotor bar. Indeed, the
detection of an incipient rotor fault remains, nowadays, still difficult.
The results obtained using the Discrete Fourier Transform phase are given in Table 3 (for example we note
H-L100 the case of a healthy rotor with 100% load and 05b-L25 the case of a partially broken rotor bar with
25% load). We represent in Figs. 8 and 10 the DFT phase when the torque applied to the machine is equal to
100% and 25% of the rated torque.
We chose to take α equal to 3. This value was selected according to the experimental results carried out.
The analysis of the results shows us that the detection of one broken rotor bar is possible at a load torque higher
than 25%. On the other hand, the detection of an incipient rotor defect (a partially broken rotor bar) can be
made only if the load torque applied to the machine is at least equal to 50%.
This non-detection is mainly due to the important noise present in the phase signal. Indeed, we can notice
that noise increases when the load applied to the machine decreases (Figs. 10(a), 10(b) and 10(c) in comparison
with Figs. 8(a), 8(b) and 8(c)). For these operating modes, the variance σn is important, which makes the
detection of the rotor defect more difficult because the ratio σc /σn is too weak so that the algorithm of decision
considers that the rotor cage does not present a defect.
When the asynchronous machine operates with no load, the algorithm of detection does not take any decision.
That means that the jump located at the frequency (1 − 2s)fs is not detectable due to its low amplitude. The
current which crosses the rotor bars is not important enough to create a consequent jump at this frequency.
Thus, the algorithm cannot evaluate the slip of the asynchronous machine and so the two frequency bands Bc
and Bn .
If we consider that an asynchronous machine operates most of its time at its rated torque, we can be satisfied
with the results given by the analyses of the Discrete Fourier Transform phase.
We show in Table 4 the results obtained with the analytical signal phase. For these analysis, we chose the
α coefficient equal to 10. This value is different for the two methods because the SNR of the Hilbert Transform
phase is greater than the DFT phase one.
We can note that the two rotor defects (a broken bar and a partially broken bar) are detected when the load
torque is contained between 25% and 100% of the rated torque. The non-detection of the jump at frequency
8
hal-00115405, version 1 - 6 Dec 2006
(1 − 2s)fs is always present when the machine is unloaded.
If we do not consider this particular case, we thus obtain better results compared to those given in Table 3.
This better detection is due to the fact that noise contained in the analytical signal phase is far less important
when the machine operates under low load torque. Figs. 11(a), 11(b) and 11(c) illustrate the use of Hilbert
Transform applied to the line current spectrum modulus compared to the Figs. 10(a), 10(b) and 10(c). The
weak noise contained in the frequency band [(1 + 2s)fs + 2δ , (1 + 4s)fs − 2δ ] gives a more sensitive detection
algorithm in case on incipient rotor defect. Indeed, in this case, the calculation of variances σc and σn is more
optimal than the previous method. Furthermore, it is obvious that the maxima are more easily detectable as
we can see in Figs. 9 and 11.
9
1.5
1.5
1
0.5
0
3
2
Phase (Rad)
2
2
Phase (Rad)
3
2.5
Phase (Rad)
3
2.5
1
0.5
0
−1
−1
No Fault
−1.5
−2
40
45
50
55
Frequency (Hz)
Fault
−1.5
−2
40
60
0
−1
−0.5
−0.5
1
45
50
Frequency (Hz)
55
−3
40
60
(b) Half-broken rotor bar
(a) Healthy rotor
Fault
−2
45
50
Frequency (Hz)
55
60
(c) One broken bar
2
2
1.5
1.5
1.5
1
0.5
0
0
−0.5
−1
−1
No Fault
−1.5
−2
40
Phase (Rad)
1
0.5
Phase (Rad)
1
0.5
−0.5
45
50
Frequency (Hz)
55
−2
40
60
0
−0.5
−1
Fault
−1.5
(a) Healthy rotor
45
50
Frequency (Hz)
55
Fault
−1.5
−2
40
60
(b) Half-broken rotor bar
45
50
Frequency (Hz)
55
60
(c) One broken bar
Figure 9: Fault detection with the analytic signal phase analysis under 100% load.
2
1.5
3
1
Phase (Rad)
Phase (Rad)
2
2
1
1.5
0.5
0
−1
Phase (Rad)
3
2.5
4
1
0.5
0
−0.5
Fault
0
−0.5
−1
−1.5
−2
−1
No Fault
−3
−4
40
45
50
Frequency (Hz)
55
−2
No Fault
−1.5
−2
40
60
(a) Healthy rotor
45
50
Frequency (Hz)
55
−2.5
−3
40
60
(b) Half-broken rotor bar
45
50
Frequency (Hz)
55
60
(c) One broken bar
Figure 10: Fault detection with the line current spectrum phase analysis under 25% load.
2
2
2
1.5
1.5
1.5
1
1
0.5
0.5
0
−0.5
−1
0
−0.5
−1
No Fault
−1.5
−2
40
Phase (Rad)
1
0.5
Phase (Rad)
Phase (Rad)
hal-00115405, version 1 - 6 Dec 2006
Phase (Rad)
Figure 8: Fault detection with the line current spectrum phase analysis under 100% load.
2
45
50
Frequency (Hz)
(a) Healthy rotor
55
−1
Fault
−1.5
60
−2
40
0
−0.5
45
50
Frequency (Hz)
55
(b) Half-broken rotor bar
Fault
−1.5
60
−2
40
45
50
Frequency (Hz)
(c) One broken bar
Figure 11: Fault detection with the analytic signal phase analysis under 25% load.
10
55
60
hal-00115405, version 1 - 6 Dec 2006
Rotor
state
H-L100
05b-L100
1b-L100
H-L75
05b-L75
1b-L75
H-L50
05b-L50
1b-L50
H-L25
05b-L25
1b-L25
H-L0
05b-L0
1b-L0
Rotor
state
H-L100
05b-L100
1b-L100
H-L75
05b-L75
1b-L75
H-L50
05b-L50
1b-L50
H-L25
05b-L25
1b-L25
H-L0
05b-L0
1b-L0
Table 3: Fault detection by
(1 − 2s)fs
σc
σn
frequency
43.49
0.0037 0.0067
44.03
0.0147 0.0008
43.28
2.6171 0.0002
45.08
0.0033 0.0034
45.74
0.3637 0.0001
45.45
0.1827 0.0049
No max detection
47.13
0.0004 0.0002
46.97
0.2113 0.0030
48.42
0.0712 0.0574
48,57
0.0004 0.0003
48.45
0.0043 0.0006
No max detection
No max detection
No max detection
DFT phase analysis
σc
No Default
σn
0.5525
19.4797
14500
0.9743
2500
37.4356
2.6286
70.4681
1.2412
1.1345
7.1024
Default
X
X
X
X
X
X
No decision
X
X
X
X
X
No decision
No decision
No decision
Table 4: Fault detection by analytic signal phase analysis
σc
(1 − 2s)fs
σc
σn
No Default Default
σn
frequency
43.45
0.0003 5.31E-05
5.3785
X
44.00
0.0038 4.49E-05 84.0466
X
43.54
0.1103 2.51E-05
4390
X
45.08
0.0003 5.13E-05
6.2159
X
45.70
0.0028 5.04E-05 54.6524
X
45.45
0.1103 3.56E-05
3100
X
46.79
0.0004 8.86E-05
4.4387
X
47.14
0.0014 3.66E-05 38.1296
X
47.01
0.0748 12.9E-05 578.6473
X
48.40
0.0002 36.9E-05
0.6299
X
48.56
0.0005 3.50E-05 14.2723
X
48.50
0.0239 35.7E-05
66.966
X
No max detection
No decision
No max detection
No decision
No max detection
No decision
The key point is that both techniques work with any knowledge of healthy rotor. But for an acceptable
response, the load must be at least 25% in order to have a minimum current in the rotor to make the phase
jump at (1 − 2s)fs distingable.
In case of limited number of samples (for example 4096 samples), the broken rotor bar can still be detected
by the analytic signal phase method. By contrast, the Discrete Fourier Transform method provides bad results
since the fault is not detected. This is due to the deficient resolution induced by the reduction of the sample
number. The HT method is insensitive to this variation thanks to its principle, but a reduction of the phase
jumps amplitude is visible. Consequently, we can conclude that a minimal number of samples is required to
make a good detection. For example, in our case, the detection of one broken rotor bar for a full load operation
is possible with 4096 samples, but at 50% load, 8192 samples are needed to make rotor fault detection.
11
6
Conclusion
The proposed methods in this article are based on the analysis of the line current spectrum. We showed that
information contained in the spectrum modulus (components at frequencies (1 ± 2ks)fs ) is also visible in the
DFT phase at the same frequencies, in the form of phase jumps. Indeed, these phase jumps also depend on the
presence or the absence of broken bars in the rotor cage. The results obtained with the DFT phase are satisfying
in case of one broken rotor bar, but the importance of the noise level prevents the detection of a partially broken
rotor bar. In order to reduce the noise in the phase signal and to make the detection more reliable, a second
method is put forward. This one uses the modulus of the line current Discrete Fourier Transform as the real
part of a new complex signal obtained by the Hilbert Transform. This method provides better results since we
detect a partially broken rotor bar for a load level equals to 25%. Besides, the Hilbert Transform method is less
sensible to number of samples variation than the DFT. This property could make easier the implantation on a
DSP.
The essential point described in this article is the detection without a priori knowledge of the healthy motor.
It is distinguished from the traditional methods which need this reference.
7
Acknowledgment
hal-00115405, version 1 - 6 Dec 2006
The authors wish to express their gratefulness to the Research Ministry and to H. Poincaré University for their
financial support in the development of the test-bed.
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